1
vote
2answers
43 views

Prove by induction a formula for $x_{k+1}=\frac{x_k}{x_k+2}$, $x_1=1$

I have a IT Maths exam coming up and I just can't figure out this question. Any help would be appreciated thanks. A sequence of integers $x_1,x_2,\dots,x_k,\dots$ is defined recursively by ...
2
votes
1answer
19 views

Clarification regarding the Josephus problem in Concrete Mathematics (Knuth, et al)

In page 9 of Concrete Mathematics, regarding the Josephus Problem, they state that "each person's number has been doubled then decreased by 1". $J(2n) = 2J(n) - 1$, for $n \ge 1$ I don't quite ...
3
votes
4answers
64 views

Elementary Proof of sum of alternating factorials

so I came across the rather interesting identity $$ \sum_{i = 0}^{\infty}\left[\sum_{j=0}^{i}\left[\frac{(-1)^j}{j!} \right] \prod_{j=0}^{i}\left[ (x-j)\right] \right] = x! $$ For positive integers ...
3
votes
2answers
104 views

Determinant involving recurrence

Evaluate $$\left| A \right| = \left| {\matrix{ {x + y} & {xy} & 0 & \cdots & \cdots & 0 \cr 1 & {x + y} & {xy} & \cdots & \cdots & 0 \cr 0 ...
0
votes
2answers
86 views

Integral of $\sin^n(x)$, recurrence relation, some properties

Practicing the manipulation of recurrence relations, I'm stuck on this : Defining $I(n)=\int_{0}^{\pi/2}sin^n(x)dx$, I got the recurrence relation $nI(n)=(n-1)I(n-2)$ for $n\ge2$. Now I'm also ...
2
votes
2answers
25 views

Solving divide and conquer recurrence

I have a recurrence $T(n)$ with only powers of two being valid as values for $n$. $$T(1) = 1$$ $$T(n) = n^2 + \frac{n}{2} - 1 + T(\frac{n}{2})$$ I tried to substitute $n=2^m$, which yields the ...
1
vote
2answers
109 views

Find an explicit formula for the recursive sequence (tips?)

Problem: A sequence is defined recursively as follows: Sk = 2k - Sk - 1, for all integers k greater than or equal to 1 S0 = 1 Use iteration to guess the explicit formula for the sequence. Use ...
2
votes
3answers
59 views

Proof by induction of a recursive sequence

I am studying CIE A levels Further Maths and I am stuck at a question from June 2002: Q The sequence of positive numbers $u_1,u_2,u_3,...$ is such that $u_1<4$ and ...
0
votes
1answer
38 views

Could someone explain me this induction.

I'm trying to understand a paper called "Diameter of Polyhedra: Limits of Abstraction" available here : http://sma.epfl.ch/~eisenbra/Publications/designs.pdf My problem is with the first two ...
0
votes
2answers
82 views

Use Bonnet recursion formula to prove by induction

Use Bonnet recursion formula: $P_{n+1}(x) = \frac{2n+1}{n+1} x P_n(x) - \frac{n}{n+1} P_{n-1}(x)$ to prove by induction 1) $P_n(1) = 1$ for all $n$ 2) $P_n(-x) = (-1)^n P_n(x)$ for all $n$ an for ...
0
votes
1answer
30 views

Inductive Proof Recursive Definition

Using this recursive Definition: $$a_{n} = \left\{\begin{matrix} 4 & n=1\\ a_{n-1}+4n-5 & n \geq 2 \end{matrix}\right.$$ I somehow have to prove using induction $$a_{n} = 2n^{2} - ...
0
votes
1answer
75 views

Cayley's formula for the number of trees(Recursion)

I'm trying to understand the proof by recurcion and induction for the Cayley's formula for the number of trees.While I'm trying to understand it there are some things that I don't get at all. -It ...
0
votes
0answers
26 views

Recurrence relation by expansion

I'm trying to find a general formula for the following recurrence relation: for n of the form 2^2^k S(n) = (rootn)(S(rootn))+n S(2) = 1 First, I let b = 2^2 just for readability ...
2
votes
2answers
62 views

Got stuck in proving monotonic increasing of a recurrence sequence

I'm stuck with the prove of the following recurrence sequence which was part of and old exam. $a_1:=\frac{1}{2}, a_{n+1}:=a_n(2-a_n)$ for $n \in \mathbb{N}$ I have to show that $0 <a_n < 1$ ...
0
votes
1answer
54 views

Using induction to prove a general form from a recurrence relation

I have the recurrence relation: $a_{n+2} = \dfrac{-a_{n}}{(n+2)(n+1)}$. I have done some work to identify that two cases emerge: one is n is positive, and one if n is negative. If n = 2m (even) ...
3
votes
1answer
128 views

Solve Recurrence Equation with Induction

Question: Given the recurrence equation for the recursive Fibonacci sequence program: $T(n) = T(n-1) + T(n-2) + b$ $T(0) = T(1) = a$ Using induction, show that $T(n) \leq f(n)$, where $f(n) = c2^n, ...
1
vote
1answer
72 views

Prove any $k\in\mathbb{N}$ can be created using sum of $a_{n}=2a_{n-1}-\left\lfloor \frac{a_{n-1}}{2}\right\rfloor ,a_{1}=1$

I need to prove that any integer can created using a sum of elements in $a_{n}=2a_{n-1}-\left\lfloor \frac{a_{n-1}}{2}\right\rfloor ,a_{1}=1$ without repetition. My attempt was just bad... I've ...
1
vote
1answer
28 views

Proving the monotonicity of a recurrence.

Define the following recurrence for $n = 1, 2, \cdots$ $T(n) = ( 1 - \operatorname{H}(\frac{1 - P^{\frac{1}{n}}}{2}))^n$ where $0 < P < 1$ is a constant, function $\operatorname{H}(\cdot)$ is ...
1
vote
1answer
55 views

Finding the explicit formula for the succession $x_0=2, x_{n+1} = 5x_n$ and proving it with induction

I'm trying to learn about recursion, first with this exercise: Find the explicit formula for the succession $$x_0=2, x_{n+1} = 5x_n$$ So, from what I've seen, I should test a bit. I see that the ...
1
vote
1answer
267 views

Combinatorics - Catalan numbers recursive function proof by induction help

I'm trying to prove that $$C_n = \sum_{i=0}^{n-1} C_iC_{n-i-1}$$ when $1\leq n$ and $C_0 = 1$ by induction. ($C_n$ is the $n$th catalan number). For $n=1$ this is true. Assume it is true for $n=k$: ...
2
votes
4answers
121 views

Recurrence sequence limit

I would like to find the limit of $$ \begin{cases} a_1=\dfrac3{4} ,\, & \\ a_{n+1}=a_{n}\dfrac{n^2+2n}{n^2+2n+1}, & n \ge 1 \end{cases} $$ I tried to use this - $\lim ...
1
vote
2answers
79 views

Induction Proof (relating two recurrences)

Let $L(n) = n + 2 L\left(\frac{n}{2}\right), \, L(1) = 1,$ and $U(n) = 9n + 2U\left(\frac{n}{2}\right), \, U(1) = 9.$ Prove by induction that $U(n) = 9L(n)$ where $n = 2^k$. I attempted to prove ...
0
votes
1answer
46 views

Solve non-linear recurrence

I am trying to solve the following recurrence (i.e. determine $a_n$). I'm not entirely sure how to proceed, I have only encountered linear ones so far. This is not a homework, I'm just doing it for ...
0
votes
2answers
44 views

Fibonacci Recursion Equation

For the Fibonacci sequence, prove the formula $a^2_{n+1} = a_n a_{n+2} + (-1)^n$ using induction. I have done the base case, when $n=3$, because for the Fibonacci sequence, $a_1=a_2=1$. I have no ...
2
votes
3answers
90 views

How do you solve a recurrence with a summation function inside

Show that $$t(n) = 1 + \sum_{ j=0}^{n-1} t(j)$$ is the same as $$t(n) = 2^n$$ Initial condition $t(0) = 1$
0
votes
3answers
227 views

Prove the Number of Additions of Fibonacci Number Algorithm

I am studying for a final exam and I'm having trouble with this question: The following recursive algorithm FIB takes as input an integer $n \ge 0$ and returns the $n$-th Fibonacci number $F_n$: ...
0
votes
1answer
62 views

Induction proof with no terms of sequence

The sequence $[x_n]$ is given by $x_1=1$ and $x_{n+1}=\displaystyle\frac{4+x_n}{1+x_n}$ for $n\ge 1$. Prove by induction that for $n\ge 1$, ...
3
votes
2answers
595 views

Inductive proof of the closed formula for the Fibonacci sequence [duplicate]

Fibonacci numbers $f(n)$ are defined recursively: $f(n) = f(n-1) +f(n-2)$ for $n > 2$ and $f(1) = 1$, $f(2) = 1$. They also admit a simple closed form: $$\sqrt 5 f( n ) = \left(\dfrac{1+ \sqrt ...
1
vote
1answer
69 views

Is this recursion well-defined?

I have a recursion defined by $$ f(n)=\max\{0,-c+pf(n-1)+(1-p)f(n+1)\} $$ with $0.5<p \leq 1$ and $f(0)=R>0$ and $f(m)=0$ for some $m>0$. I am trying to show that $f(n)$ is decreasing in ...
0
votes
3answers
77 views

What are the coefficients of the polynomial inductively defined as $f_1=(x-2)^2\,\,\,;\,\,\,f_{n+1}=(f_n-2)^2$?

Let $\{f_n(x)\}_{n\in \Bbb N}$ be a sequence of polynomials defined inductively as $$\begin{matrix} f_1(x) & = & (x-2)^2 & \\ f_{n+1}(x)& = & (f_n(x)-2)^2, &\text{ for all ...
0
votes
1answer
222 views

$n$ Lines in the Plane

How am I to "[u]se induction to show that $n$ straight lines in the plane divide the plane into $\frac{n^2+n+2}{2}$ regions"? It is assumed here that no two lines are parallel and that no three lines ...
1
vote
3answers
133 views

Using complete induction, prove that if $a_1=2$, $a_2=4$, and $a_{n+2}=5a_{n+1}-6a_n$, then $a_n=2^n$

Could anyone please explain to me how to do this problem by using the principle of complete induction? Thanks. :) Let $a_1=2$, $a_2=4$, and $a_{n+2}=5a_{n+1}-6a_n$ for all $n\geq 1$. Prove that ...
1
vote
1answer
54 views

Prove by induction that $d_n=2^n+3^n$, where $d_n = 5d_{n-1}-6d_{n-2}$

I have one more induction question. $d_0 =2 $ $d_1=5$ let $d_n=5d_{n-1} - 6d_{n-2}$ Prove that $d_n=2^n+3^n$
0
votes
2answers
32 views

recurrence relation with induction

The following recurrence relation: $T(n)=T(n-1)+n=1+\frac{n^2+n}{2}=\theta(n^2)$, so this mean that: there is $c_1, c_2, n_o > 0 : c_1n^2<=1+\frac{n^2+n}{2}<=c_2n^2$, the second inequality ...
3
votes
2answers
63 views

Soving Recurrence Relation

I have this relation $u_{n+1}=\frac{1}{3}u_{n} + 4$ and I need to express the general term $u_{n}$ in terms of $n$ and $u_{0}$. With partial sums I found this relation $u_{n}=\frac{1}{3^n}u_{0} + ...
1
vote
1answer
173 views

Generalized Josephus problem

I have been reading generalized Josephus problem from Concrete Mathematics. The recurrence form for the problem is given as f(1) = a f(2n) = 2f(n) + b, for n >= 1 f(2n+1) = 2f(n) + y, for n >= 1 ...
0
votes
1answer
343 views

Solving Recurrence Relation with Forward Substitution

I've found myself quite stuck on this recurrence relation. I've been given it to solve, via forward substitution and verify using induction. I start out with $$ T(n) = 4T(n/3) $$ For all $n > 1$ ...
0
votes
1answer
167 views

Induction hypothesis when proving solution to linear homogeneous recurrence equation

I am looking at an example solution to a linear homogeneous recurrence equation of: $T(0) = 0$ $T(1) = 2$ $T(n) = 4T(n-1) - 3T(n-2), n > 1$ And solving it you get $T(n) = 3^n - 1$ In the ...
0
votes
3answers
74 views

Recursive/Fibonacci Induction [duplicate]

1) Let $F_n$ denote the $n^t$$^h$ Fibonacci number. Prove by induction: $$ F_n = \frac{\left(\frac{1+\sqrt{5}}{2}\right)^{n+1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n+1}}{\sqrt{5}} $$ Clear ...
1
vote
2answers
186 views

Difference between two methods of induction for proving the correctness of recurrence equation solution

Suppose you have the recurrence equation $T(0) = 0$ $T(n) = 2T(n-1) + 1, n > 0$ The closed form of this equation appears to be $T(n) = 2^n - 1$ To prove this is correct using induction, we have ...
0
votes
4answers
361 views

Having a lot of trouble solving this recurrence with iteration and finding a closed form…

I'm learning discrete math and didn't have any trouble with any recurrences in the examples I went over through the chapters on it, but this one problem at the end of the first chapter is killing me, ...
0
votes
0answers
63 views

proving an inequality by induction

Not sure how to proceed. I'm trying to prove that the following inequality is true. I know that $t_2 = 6$ and $t_3=17$ from the problem statement. The base case is obvious. $t_{r+1} \leq (r+1) (t_r ...
1
vote
1answer
76 views

Prove by Induction $\frac{a_n-\sqrt{A}}{a_n+\sqrt{A}} = \left[\frac{a_1-\sqrt{A}}{a_1+\sqrt{A}}\right]^{2^{n-1}} $

$a_1=\frac{1}{2}(a_0+\frac{A}{a_0})$; $a_2=\frac{1}{2}(a_1+\frac{A}{a_1})$; and $a_{n+1}=\frac{1}{2}(a_n+\frac{A}{a_n})$ for $n \geq 2$; where $a\gt 0$, $A\gt 0$. Prove: ...
2
votes
1answer
516 views

Concrete Mathematics - The Josephus Problem

I've been going through Concrete Mathematics and have a question on the inductive proof for the Josephus problem. Recurrent relation: $J(1) = 1$ $J(2n) = 2J(n) - 1$ $J(2n+1) = 2J(n) + ...
0
votes
1answer
41 views

How can I prove a sequence of values follows a certain closed form equation?

For example, imagine I'm trying to do this $$ (1-3x+3x^2)/(1-3x+3x^2-3x^6) = \sum\limits_{n=0}(a_nx^n) $$ $$ (1-3x+3x^2) = \sum(a_nx^n) * (1-3x+3x^2-3x^6) $$ Then say we are given some closed ...
13
votes
5answers
392 views

Alternate proof that for every natural number $n,\ \left\lfloor\left(\frac{7+\sqrt{37}}{2}\right)^n\right\rfloor$ is divisible by $3$

Original Problem: Prove that for every natural number $n$,$$\left\lfloor\left(\frac{7+\sqrt{37}}{2}\right)^n\right\rfloor$$ is divisible by $3$. I found the problem in the book Winning ...
2
votes
3answers
138 views

Mathematical induction proof; $g_k=3g_{k-1} - 2g_{k-2}$

Can someone help me with this problem? I'm having a hard time proving this. It's been a long time since I have done mathematical proofs. Suppose that $g_1,\ g_2,\ g_3,\ \ldots$ is a sequence of ...
0
votes
1answer
131 views

Proving the equality of 2 functions

You are given that $f(n)=g(n, f(n-1))$ (some initial values given). Looking at the first few terms, it becomes obvious that $f(n)=h(n)$. How does one go about proving this? Induction seemed obvious ...
2
votes
1answer
63 views

Flawed twofold induction on an inequation (but where?)

The following induction is flawed because the result admits a counter-example, but I can't find where is the flaw. Please advise. [Edit: As pointed out in the answer, the error was in the base case. ...
0
votes
1answer
102 views

Demonstration of a recurence formula

I would like to demonstrate a recursive formula that I have inferred. My background in mathematics is not quite as high as I hoped and although I have tried to apply the basic tricks that go with the ...